Experimental metrology to obtain thermal phonon transmission coefficients at solid interfaces

Interfaces play an essential role in phonon-mediated heat conduction in solids, impacting applications ranging from thermoelectric waste heat recovery to heat dissipation in electronics. From the microscopic perspective, interfacial phonon transport is described by transmission coefficients that link vibrational modes in the materials composing the interface. However, direct experimental determination of these coefficients is challenging because most experiments provide a mode-averaged interface conductance that obscures the microscopic detail. Here, we report a metrology to extract thermal phonon transmission coefficients at solid interfaces using ab initio phonon transport modeling and a thermal characterization technique, time-domain thermoreflectance. In combination with transmission electron microscopy characterization of the interface, our approach allows us to link the atomic structure of an interface to the spectral content of the heat crossing it. Our work provides a useful perspective on the microscopic processes governing interfacial heat conduction.

Figure 1

(a) Schematic of the principle underlying the measurement of transmission coefficients. If the characteristic length scale of the thermal transport is much longer than the phonon MFPs, information about the interfacial distribution is lost due to strong scattering. If some MFPs are comparable to the thermal length scale, the nonequilibrium distribution at the interface propagates into the substrate where it can be detected. (b) 2D schematic of the experimental configuration subject to a modulated heating source: a double layer structure of a transducer film on a substrate (sample). $Q_0$ is the amplitude of the heating source, $\eta$ is angular modulation frequency, $\delta$ is the optical penetration depth of the heating source, and $x$ is the cross-plane transport direction. $x_1$ and $x_2$ correspond to the coordinate systems used in transducer and substrate accordingly.

Figure 2

(a) Two artificial transmission coefficient profiles from Si to Al for the longitudinal branch. The two profiles have different trend: a constant value (dashed line) and an increasing trend (dotted-dash line), but both give the same interface conductance. (b) The amplitude of the simulated TDTR signals using a constant transmission coefficient (dashed line) and a increasing transmission coefficient profile (dotted-dash line) with silicon's dispersion and a cutoff scattering rates such that there no MFP exceeding 50 nm. (c) The amplitude of the measured TDTR signal (solid line) for Al/Si at 300 K and the simulated TDTR signals using the transmission coefficient profiles shown in (a). (Insets) The phase of the corresponding TDTR signals. The two signals are almost identical under diffusion transport, while, in the quasiballistic regime, the two simulated TDTR signals are no longer identical and do not match with the measured signal either.

Figure 3

Experimental TDTR data (symbols) on this sample at $T=300$ K for modulation frequencies $f=2.68$, 5.51, and 9.79 MHz along with the (a) amplitude and (b) phase fit to the data from the BTE simulations (shaded regions), demonstrating excellent agreement between simulation and experiment. The shaded stripes denoted BTE simulations correspond to the likelihood of the measured transmission coefficients possessing a certain value as plotted. Inset: TEM image showing the clean interface of an Al/Si sample with the native oxide removed. The interface thickness is less than 0.5 nm. Transmission coefficients of longitudinal phonons ${\mathrm{T}}_{\text{Si}\to \text{Al}}\left(\omega \right)$ (blue shaded region) vs (c) phonon frequency and (d) phonon wavelength, along with the DMM transmission coefficient profile (green dashed line) that gives the same interface conductance as the measured ${\mathrm{T}}_{\text{Si}\to \text{Al}}\left(\omega \right)$. The intensity of the shaded region corresponds to the likelihood that the transmission coefficient possesses a given value.

Figure 4

(a) Optimized transmission coefficient profile (dashed line) and the profile with 90% reduced values at phonon frequencies less than 6 THz (solid line) vs phonon frequency for longitudinal modes at a clean Al/Si interface. (b) The absolute phase difference between the TDTR signal and experimental data as a function of time using the optimized transmission coefficient profile (dashed line) and the profile with 90% reduced values at low phonon frequencies (dotted-dash line) compared to the experimental uncertainties (solid line).

Figure 5

Experimental TDTR data (symbols) on Al/Si with a clean interface at 300 K for modulation frequency $f=9.79$ MHz along with the (a) amplitude and (b) phase compared to the data from the BTE simulations using constant ${\mathrm{T}}_{\mathrm{Si}\to \mathrm{Al}}$ (dash-dotted lines) and DMM (dotted lines). (c) Amplitude and (d) phase difference between averaged experimental data and the BTE simulations using constant ${\mathrm{T}}_{\mathrm{Si}\to \mathrm{Al}}$ (dash-dotted lines), DMM (dotted lines), and the optimal profile in Fig. 2 of the main text (dashed lines). The solid line indicates the uncertainty in experiments.

Figure 6

Spectral heat flux with the measured (blue shaded region) and DMM (green dashed line) transmission coefficient profiles across the interface versus (a) phonon frequency and (b) phonon wavelength. Phonons with frequencies less than approximately 4 THz carry a significant amount of heat across the interface. The intensity of the shaded region reflects the likelihood of the corresponding transmission coefficients.

Figure 7

(a) Amplitude and (b) phase as a function of time at modulation frequencies $f=2.99$, 5.17, and 9.79 MHz from experiments (symbols) and simulations (shaded regions) for Al on Si with a clean interface at 400 K. (c) Amplitude and (d) phase as a function of time at modulation frequencies $f=2.9$, 5.3, and 9.8 MHz from experiments (symbols) and simulations (shaded regions) for Al on SiGe with a clean interface at 300 K. The magnitude and trend of the experimental data are reproduced using the same transmission coefficient profile as in Fig. 3 without any adjustable parameters.

Figure 8

TEM images showing the Al/Si sample with a (a) native oxide layer (thickness $\sim 1$ nm) and a (b) thermally grown oxide layer (thickness $\sim 3.5$ nm). (c) Amplitude of the surface temperature decay curves at modulation frequencies $f=2.1$, 5.10, and 9.80 MHz of experiments (symbols) and simulations (shaded regions) for Al on (c) Si with a native oxide layer. (d) Amplitude of the surface temperature decay curves at modulation frequencies $f=2.1$, 5.3, and 14.5 MHz of experiments (symbols) and simulations (shaded regions) for Al on Si with a thermally grown oxide layer. Insets: corresponding phase of the surface temperature decay curves. The corresponding transmission coefficient profiles vs (e) phonon frequency and (f) phonon wavelength show that as the interface gets rougher, phonons with frequencies less than 4 THz are more likely to be reflected.

Figure 9

Calculated transient surface temperature (a) amplitude and (b) phase for Al on Si using a 1D (solid blue lines) and 3D (dashed red lines) BTE model. The cross-plane transport completely dominates the thermal process with the pump size we used in the experiments.

Figure 10

The surface temperature decay subject to a surface impulse heating for Al on Si with (solid blue line) and without (dashed red line) the effects of electrons. After 100 ps, the heat is dominated by phonons and there is little contribution from the electrons. Therefore the electrons have negligible effects on the signal on the time scale relevant to the heat conduction across interfaces.

Figure 11

TDTR surface temperature decay curves along with (a) amplitude and (c) phase using different transmission coefficient profiles with two completely random partitions of transmitting modes to different polarizations (dashed and dash-dotted lines) and without mode conversion (solid line; assuming phonons maintain their polarization as they cross the interface). (c) Spectral interfacial heat flux vs phonon frequency predicted using different transmission coefficient profiles with mode conversion (dashed and dash-dotted lines) and without mode conversion. Mode conversion does not have an effect on the signal and the conclusions beyond the uncertainties already considered in our model.

Figure 12

Transmission coefficients from Si to Al vs (a)–(c) phonon frequency and (d)–(f) phonon wavelength for different polarizations measured from Al/Si sample with three different interfaces studied in this work. The intensity of the shaded region corresponds to the likelihood that the transmission coefficient possesses a given value. We emphasize that the transmission coefficients for the three polarizations are the only fitting parameters in our calculations.

Figure 13

Transmission coefficients from Al to Si vs (a)–(c) phonon frequency and (d)–(f) phonon wavelength for different polarizations measured from Al/Si sample with three different interfaces studied in this work, calculated using the measured values of the transmission coefficients from Si to Al shown in Fig. 12. The increase in transmission coefficient from Al to Si at phonon frequencies less than approximately 4 THz is due to the requirements of detailed balance. Specifically, these coefficients must follow the shape of the density of states since the coefficients from Si to Al are a constant value. These coefficients are determined by the principle of detailed balance and are not free parameters.

Figure 14

Experimental TDTR data (symbols) of an Al/Si sample with a clean interface at $T=300$, 350, and 400 K at different modulation frequencies fit to the data from the BTE simulations (shaded regions), demonstrating excellent agreement between simulation and experiment at different temperatures.

Figure 15

Experimental TDTR data (symbols) of an Al/Si sample with a native oxidized interface, an Al/Si sample with a thermally oxidized interface, and an Al/SiGe with a clean interface at $T=300$ K at different modulation frequencies fit to the data from the BTE simulations (shaded regions).

Figure 16

Calculated transient surface temperature (a) amplitude and (b) phase for Al on Si using a two-layer diffusive model with Al thermal conductivity to be 230 (solid blue line) and 123 $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ (dash-dotted red line). The surface temperature response is not sensitive to the change of Al thermal conductivity from 230 to 123 $\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$.